Is it correct that every distribution will eventually converge to the normal distribution? Or is it...
Is it correct that every distribution will eventually converge to the normal distribution? Or is it only the distribution of unbiased/iid events? For example, if we toss two biased dice for an infinite amount of time (biased to only give the combined outcome of 11 or some other number), will the distribution of its outcomes then converge to the uniform distribution or to the normal distribution?
Homework Answers
No, It is not correct that every distribution will eventually
converge to normal distribution, even Unbiasedness and IId events
are sufficient conditions for the CLT( Central Limit Theorem
).
Second sufficient condition for CLT to exist is presence of second
order moment.
Now for Cauchy distribution moments do not exist hence CLT will not
be followed by Cauchy distribution.
According to your example we do not know the amount of bias
present there,even then distribution will not follow normal as it
is violating the sufficient condition for
CLT.
Central Limit
Theorem
If Xi (i=1,2,3..,n) be independent random variables
such that E[Xi] = μi and
V[Xi] = σi2 : then under
certain very general conditions, the the random variable
Sn = X1 + X2 + .... +
Xn is asymptotically normal
with mean μ and standard deviation
σ where
μ = Σμi and σ2 =Σσi2
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some situations, this choice of d.f. may increase
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slightly more "conservative" answer.
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